On a certain planet, it takes a village to beget a child. Well, not an entire village in fact, but instead of a father offering sperm and a mother providing an egg, five parents must get together to produced a fertilized egg. The egg will produce 1,000 children, 200 of each of the five sexes. Your job is to figure out which sexes are genetically important in determining an offspring’s (non sex-related) traits.

It may help you to know the history of genetics on earth. Even Aristotle and Hippocrates realized that human traits could be inherited. It took a now-famous but then-obscure monk, Gregor Johann Mendel, to perform crosses and recrosses among different strains of peas to understand what was going on. Mendel knew that a single grain of pollen could fertilize a pea plant’s ovum. He assumed that each character trait came from some combination of the mother and the father. The question was how.

The simplest model is that each possible trait has two possible values and children take one or the other. Suppose the pollen comes from the male of a plant strain having a value B for some trait, whereas the female plant strain has trait value b. Then about half the children would have trait value B and half would have trait value b.

But Mendel observed instead that they all looked as if they had trait value B. In modern parlance, they had the B phenotype. But then he observed that if he crossed the males and females of the children, about a quarter of them had the b phenotype. That is, the grandchildren have more variety than the children! He showed that this occurred whether the original B plants were male or female.

What was going on? Mendel reasoned that children take values from both parents but that one value was "dominant" and one "recessive." That is, the original parents are either BB or bb. The children are all Bb (taking one value from the father and one from the mother). But, Mendel reasoned, if B were dominant over b, then the Bb children would all look like BB parents. That is, they would have the B phenotype.

In the second generation, a Bb father and a Bb mother will produce approximately even numbers of children that are BB, Bb, bB and bb. A quarter of them would have BB traits, a quarter would have bb traits, and half would have Bb or bB traits (which are indistinguishable). That is in fact what Mendel found (within statistical error).

On our alien planet, five sexes (S1, S2, S3, S4 and S5) are involved although we don't know whether all five contribute genetically to the offspring. (Some of the sexes may transfer sperm or eggs between their co-mates, or may just carry the developing offspring.) You have located one strain, all of whose individuals have phenotype B (even after lots of crossing), and another strain having all individuals with value b.

In the first breeding experiment, S1, S2 and S3 all have value B (and because of the crossing experiments, can be assumed to be pure B’s) whereas S4 and S5 have value b. All 1,000 children in the next generation have the B phenotype. But in the following generation about 74% have phenotype B and 26% have the b phenotype.

In the second breeding experiment, if S1, S2 and S4 are pure B whereas S3 and S5 are pure b, then all 1,000 children in the next generation have the B phenotype. In the following generation again about 74% have the B phenotype and 26% have the b phenotype.

Problem 1: What can you tell from the information so far?

Statistically, the 74% is significantly different from the 75%, but you still suspect that maybe just two sexes are important. However, when you cross all 26% of those that have the b phenotype, more than 15% of their children have the B phenotype.

Problem 3: Now suppose you do a third breeding experiment in which you start with S1, S2 and S5 having the value B whereas S3 and S4 have the value b, then all 1,000 children in the next generation have the b phenotype. This is different! In the following generation, 26% have the B phenotype and 74% have the b phenotype. Crossing the B phenotype children gives more than 15% that have the b phenotype – symmetric to the other case. Can you make a good hypothesis about which parents determine the trait value and what the rule is for which trait will show up?

ABOUT THE AUTHOR(S)

Dennis Shasha is at the Courant Institute of Mathematical Sciences, New York University. His most recent puzzle book, Puzzles for Programmers and Pros, was published in May 2007 by John Wiley and Sons/Wrox.

Scientific American is part of Springer Nature, which owns or has commercial relations with thousands of scientific publications (many of them can be found at www.springernature.com/us). Scientific American maintains a strict policy of editorial independence in reporting developments in science to our readers.